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Research ArticleRobust Sliding Mode Control Based on GA Optimization andCMAC Compensation for Lower Limb Exoskeleton

Yi Long, Zhi-jiang Du, Wei-dong Wang, and Wei Dong

State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Wei Dong; [email protected]

Received 16 January 2016; Accepted 17 February 2016

Academic Editor: Huapeng Wu

Copyright © 2016 Yi Long et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A lower limb assistive exoskeleton is designed to help operators walk or carry payloads. The exoskeleton is required to shadowhuman motion intent accurately and compliantly to prevent incoordination. If the user’s intention is estimated accurately, aprecise position control strategy will improve collaboration between the user and the exoskeleton. In this paper, a hybrid positioncontrol scheme, combining sliding mode control (SMC) with a cerebellar model articulation controller (CMAC) neural network,is proposed to control the exoskeleton to react appropriately to human motion intent. A genetic algorithm (GA) is utilized todetermine the optimal sliding surface and the sliding control law to improve performance of SMC. The proposed control strategy(SMC GA CMAC) is compared with three other types of approaches, that is, conventional SMC without optimization, optimalSMC with GA (SMC GA), and SMC with CMAC compensation (SMC CMAC), all of which are employed to track the desiredjoint angular position which is deduced from Clinical Gait Analysis (CGA) data. Position tracking performance is investigatedwith cosimulation using ADAMS and MATLAB/SIMULINK in two cases, of which the first case is without disturbances while thesecond case is with a bounded disturbance. The cosimulation results show the effectiveness of the proposed control strategy whichcan be employed in similar exoskeleton systems.

1. Introduction

The lower extremity exoskeleton, which began in the late1960s, is an electromechanical structure worn by humanusers as an intelligent device for performance assistance andenhancement. In recent years, wearable robots have attractedinterests of many researchers widely. The Berkeley LowerExtremity Exoskeleton (BLEEX)was designed to assist peoplein walking for carrying load, which could walk at the speedof 0.9m/s while carrying 34 kg payload [1]. A mechanical leghas seven DOFs (three at the hip, one at the knee, and threeat the ankle), of which four DOFs are actuated by valve-basedhydraulic actuation systems [2]. However, these many activeDOFs make the system complex and heavy, weighing 38 kg.The latter exoskeletons, that is, ExoHiker, ExoClimber, andHULC, simplifymechanical structure and reduce the numberof active DOFs while carrying more payloads up to 68 kg–90 kg [3]. Hybrid Assistive Limb (HAL), proposed by theUniversity of Tsukuba in Japan, has two active DOFs at thehip joint and knee joint, which are controlled according to

collected electrical signals from muscles [4]. HAL is used tohelp users carry load and assist disabled people in walking[5, 6]. An underactuated exoskeleton system is designedbased on appropriate criteria to help infantry soldiers walkon different terrain, where active joints are applied to the kneejoints while other joints are passive [7]. Moreno et al. studiedand analyzed the human interactionwithwearable lower limbexoskeleton, where the robot gathered information from thesensors in order to detect human actions and subjects alsomodified their gait patterns to obtain the desired responsesfrom the exoskeleton [8].

Although many kinds of lower limb exoskeleton robotsare studied, the human-exoskeleton collaborative movementis quite complex and difficult due to nonlinear character-istics of dynamic model and uncertainties, for example,external disturbance and involuntary movements. To achievethe goal of making exoskeletons providing assistance forhuman beings, a consistent dynamic tracking performance isrequired to maneuver exoskeletons in an efficient, smooth,and continuous manner [9]. The control procedure can be

Hindawi Publishing CorporationApplied Bionics and BiomechanicsVolume 2016, Article ID 5017381, 13 pageshttp://dx.doi.org/10.1155/2016/5017381

http://dx.doi.org/10.1155/2016/5017381

2 Applied Bionics and Biomechanics

divided into two steps, acquiring human motion intent withhuman-robot interaction (HRI) and following the humanmotion intent accurately.

When the wearer wants to move, the central controllersends control signals to enforce the exoskeleton to followcommanded signals, during which HRI decreases. A crucialissue of control is to follow the estimated human motionintent accurately. The more accurate the intention trackingis, the more compliantly the exoskeleton works. The pre-cise motion control of robotic manipulators has receivedconsiderable attention from many robotics researchers andits challenges continue to limit overall control performancebecause of structured and unstructured uncertainties [10].In exoskeletons, the structured uncertainties contain payloadvariations, while unstructured uncertainties contain sensornoises, joint friction, and external disturbances. There aremany approaches for position control approaches to deal withuncertainties such as robust control [11, 12], adaptive control[13, 14], intelligent control [15], and sliding mode control[16].

SMC is a robust control approach that drives state tra-jectory to predefined sliding surface by using discontinuouscontrol inputs [17], which is used to improve control per-formance for robotic manipulators with model uncertaintiessuch as parameter perturbations, unknown joint frictionsand inertias, and external disturbances [18]. It is notable thatits overall performance is superior to general PID controlalgorithm [19]. The process of designing a SMC controllerhas two steps: defining suitable sliding surfaces and designingdiscontinuous control laws [13]. Parameters of SMC shouldbe chosen suitably to obtain optimal performance. Somecommon optimization methods are provided and applied inrobots, for example, GA [20], particle swarm optimization(PSO) [21], ant colony optimization (ACO) [22], and evolu-tionary algorithm (EA) [23]. GA is simple to be implementedand is capable of locating global optimal solutions [24], whichis utilized to optimize the structure of intelligent methods[25, 26]. The decoupled SMC as a supervisory controller isapplied in accordance with PID control, whose parametersare tuned using GA, to enhance tracking performance andeliminate the chattering problem [27]. The gain switch andsliding surface constant parameters are selected byGA so thatthe designed SMC can achieve satisfactory performance [28].However, GA is only used to optimize parameters of slidingsurfaces or SMC control laws. In this work, we use GA tooptimize all parameters of the sliding surface and the controllaw at the same time.

The optimal SMC can deal with uncertainties to achievesatisfactory performance. To improve tracking performance,CMAC is added as a compensation item with property offast learning capability. The CMAC proposed first by Albus[29] is similar to the mode of human cerebellum, whichis an autoassociative memory feed-forward neural network.Compared with other feed-forward neural networks, it hasfaster convergence speed [30]. The approach which usesCMAC as a compensation item with SMC is applied inposition control of robotic manipulators [31]. In this work,we propose to combine optimal SMC using GA and CMACcompensation to form the hybrid position control strategy.

Connect to backpack

Hydraulic actuator

Sensor

Active joint

Passive joint Wearable shoes

𝜃hip

𝜃knee

Figure 1: Prototype of lower limb powered exoskeleton. Thereare two active joints of each leg in walking direction, which arerepresented as 𝜃hip and 𝜃knee. All auxiliary facilities are packaged inthe backpack.

The remainder of this paper is organized as follows. Thespecific system under study is given in the second section.In Section 3, the proposed control strategy is explainedin details. Cosimulations using the proposed approach andresults analysis are presented in the fourth section. Conclu-sions are drawn in the final section.

2. Problem Formulations

2.1. Exoskeleton Configuration. Based on principles in bio-logical design, the designed exoskeleton is required to retainadaptability to multifunctionality of human lower limbs. Anavailable powerful tool when designing an assistive exoskele-ton is the enormous Clinical Gait Analysis (CGA) data onhuman walking [32]. With CGA data [33], our designedexoskeleton is shown in Figure 1. As Figure 1 shows, thereare two active joints of a single leg in sagittal plane, whichare knee joint and hip joint actuated by hydraulic actuationsystem.

2.2. Mathematical Model of Exoskeleton. For multirigid sys-tem, Euler-Lagrange is a frequently used method for mod-eling of robotic manipulators. The exoskeleton is a typicalhuman-robot collaboration system, which includes the user’slower limbs and mechanical limbs which are tied together atthe interaction cuffs. Mathematical model of a single leg ofexoskeleton is obtained because of its symmetry structure.Without loss of generality, the dynamic equation of the swingleg of exoskeleton robot can be expressed as follows:

M (q) q + C (q, q) q + G (q) +D = T, (1)

whereM(q) ∈ 𝑅𝑛×𝑛 is the symmetric definite inertial matrix;C(q, q) ∈ 𝑅

𝑛×𝑛 is the Coriolis and centrifugal force matrix;G(q) ∈ 𝑅

𝑛×1 is the gravitational force matrix; T ∈ 𝑅𝑛×1

Applied Bionics and Biomechanics 3

is the control input vector; D ∈ 𝑅𝑛×1 denotes unmodeled

dynamics and external disturbances.For dynamics model in (1), several properties are pre-

sented as the following [34].

Property 1. MatrixM(q) is symmetric and positive definite.

Property 2. Matrix M(q) − 2C(𝑞, ��) is a skew-symmetricmatrix if ∀𝜀 ∈ 𝑅𝑛, 𝜀𝑇(M(q) − 2C(𝑞, ��))𝜀 = 0.

Property 3. There exist finite scalars 𝛿𝑖> 0, 𝑖 = 1, . . . , 4 such

that ‖M(q)‖ ≤ 𝛿1, ‖C(q, q)‖ ≤ 𝛿

2, ‖𝐺(q)‖ ≤ 𝛿

3, and ‖𝐷‖ ≤

𝛿4, which means all items in dynamic model are bounded.

In the position control of robotic manipulators, we definetrajectory tracking error as

e = q𝑑− q, (2)

where e is the tracking error, q𝑑is reference trajectory, and q

is actual trajectory. Based on (2), we can obtain

e = q𝑑− q,

e = q𝑑− q,

(3)

where e and e is the first and second derivative of e, q𝑑and

q𝑑are angular velocity and acceleration vector of command

input, and q and q are that of actual output, respectively, allof which are bounded.

3. Control Strategy Design

3.1. Sliding Mode Control. A general SMC design consistsof two steps: the sliding surface design and the controllaw construction. The purpose of the SMC is to track thetrajectory specified by human intention and maintain systemtrajectory in the sliding surfaces [18]. Considering that thereexist uncertainties including unmodeled frictions, variationof parameters, and external disturbances, the robustnessshould be an important concern in the controller design forexoskeleton system. The general sliding surface is defined ass = e + Ae. To improve robustness of controller, a designedintegral sliding surface is represented as follows [35]:

s = e + Ae +H∫

𝑡𝑠

0

e 𝑑𝑡, (4)

where 𝐴 and 𝐻 are positive definite matrix. Then s can bederived:

s = e + Ae +He. (5)

As the second design stage of SMC, the control laws shouldbe chosen, which should be satisfied with the existencecondition of SMC [36]:

s𝑇 s < 0. (6)

For the exoskeleton system under study, we define the SMCcontrol law as follows:

u = M (𝑞) q𝑑− (T𝑑− C (𝑞, ��) q) +M (𝑞)Ae

+M (𝑞)He + C (𝑞, ��) s + 𝜀 sgn (s) + Ks,(7)

where T𝑑= D − G(q), 𝜀 and K are positive definite matrices,

and sgn(s) is a symbolic function which is shown as follows:

sgn (s) ={{{{

{{{{

{

1, s > 0,

0, s = 0,

−1, s < 0.

(8)

The SMC algorithm has chattering phenomena, whichaffects the accuracy of position control much. In order toeliminate chattering, the continuous function 𝜃(s) with relaycharacteristics is used to replace the function of symbolicfunction sgn(s) to restrict the trajectory in a boundary layerof ideal sliding mode [37]. Then (7) can be rewritten as

u = M (𝑞) q𝑑− (T𝑑− C (𝑞, ��) q) +M (𝑞)Ae

+M (𝑞)He + C (𝑞, ��) s + 𝜀𝜃 (s) + Ks,(9)

where 𝜃(s) = s/(‖s‖ + 𝜎), 𝜎 > 0. Before stability analysis,Barbalat lemma is shown as the following [38].

Barbalat Lemma. If a differentiable function 𝑓(𝑡) has a limitas 𝑡 → ∞, and if ��(𝑡) is uniformly continuous, then 𝑓(𝑡) → 0

as 𝑡 → ∞.

Theorem 1. The proposed controller (9) guarantees asymptoticconvergence to zero, both of the trajectory tracking errors andsliding surfaces. Namely, the system is globally stable; that is,when 𝑡 → ∞, 𝑒 → 0, 𝑠 → 0.

Proof. Lyapunov function is defined as

V =1

2s𝑇M (𝑞) s. (10)

Differentiating V with respect to time yields

V = s𝑇M (𝑞) s + 1

2s𝑇M (𝑞) s. (11)

Considering Property 2, then

s𝑇 (12M (𝑞) − C (𝑞, ��)) s = 0. (12)

Combining (10)–(12), one can get

V = s𝑇 (M (𝑞) s + C (𝑞, ��) s) = s𝑇 (M (𝑞) (q𝑑− q)

+M (𝑞)Ae +M (𝑞)𝐻𝑒 + C (𝑞, ��) s) .(13)

And q can be solved by

q = M (𝑞)−1

(T + T𝑑− C (𝑞, ��) q) . (14)

Substituting (14) into (13), then

V = s𝑇 (M (𝑞) q𝑑− (T + T

𝑑− C (𝑞, ��) q) +M (𝑞)Ae

+M (𝑞)He + C (𝑞, ��) s) .(15)


Substituting (7) into (15), we can obtain

V = s𝑇 (−𝜀𝜃 (s) − Ks) . (16)

It is easy to know that K and 𝜀 are positive definite matrices;therefore s𝑇Ks > 0, s𝑇𝜃(s) > 0; then V < 0. Hence, thesystem is globally stable. With the Barbalat lemma, s → 0

as 𝑡 → ∞; then one knows e → 0 and e → 0 as 𝑡 → ∞.This control law could realize convergence of the trajectorytracking error to zero.

3.2. Genetic Algorithm. In SMC, those constant parametersexisting in sliding surfaces and control laws, which areA, H, K, and 𝜀 in (9), determine the overall performance.Hence, it is necessary to find the optimal values of themusing optimization algorithm. GA is an adaptive heuristicsearch algorithm that mimics the process of natural selectionand uses biological evolution to develop a series of searchspace points toward an optimal solution. There are five com-ponents that are required to implement GA: representation,initialization, fitness function, genetic operators, and geneticparameters [39].

A simple GA involves three types of operator: selection,crossover, and mutation [40]. Selection is a probabilisticprocess for selecting chromosomes in the population usingtheir fitness values.The chromosome with larger fitness valueis likely to be selected to reproduce. Crossover is the processof randomly choosing a locus and swaps the characters eitherleft or right of this locus between two chromosomes to createtwo offspring. The probability of crossover occurring for theparent chromosomes is usually set to a large value (e.g., 0.8).Mutation is to randomly flip some of the bits by changing “0”to “1” or vice versa, with a small probability (e.g., 0.001) whichmaintains genetic diversity to guarantee that GA can come tobetter solution. The process of GA optimization is shown inFigure 2. As Figure 2 shows, there are parameters such as thesize of population and generation and the length of code thatshould be initialized; then the process of selection, crossover,and mutation is preceded until the convergence conditionsare satisfied.

3.3. SMC with GA Optimization. Based on that discussedabove, the fitness function should be confirmed beforeimplementing GA to SMC. The goal of SMC is to achieveprecise trajectory tracking for robotic manipulators; that is,the smaller the trajectory errors are, the more effective thecontroller is. Those parameters to be optimized are relevantto trajectory error; hence the fitness function is defined asfollows:

𝐽 (A,H,K, 𝜀) =∞

∑

𝑘=0

‖e (𝑘)‖2 . (17)

With the fitness function, those parameters can be foundwith theminimization of tracking errors during the trajectorytracking using the designed control law. In the search spaceof GA, the SMC will have optimal parameters when thefitness function has minimum values. The algorithm of SMCoptimized by GA is shown as Algorithm 1 in Appendix A.

Stop

Outputoptimal parameters

Selection

Crossover

Mutation

Start

Replace old populationwith a new one

YesNo

Fitness evaluation

Converaged

Define the size of population, generation,

and code length

Set the range of parameters

Initialpopulation

Fitness value

Figure 2: The process of GA optimization, when the convergencecondition is satisfied, the optimal parameters will be obtained.

InputState space

Association cell Memory cell

Sum

Output∑

Figure 3: Structure of CMAC neural network.

3.4. CMAC Neural Network. The CMAC neural network hasthree steps: projecting an input into association area, com-pressing memory cell through Hash coding, and calculatingthe output as a scalar product of the memory area [41], whichis shown in Figure 3. The output of CMAC can be expressedas follows [42]:

y𝑠= C𝑇𝑠HW

= [𝑐𝑠,1

𝑐𝑠,2

⋅ ⋅ ⋅ 𝑐𝑠,𝑁ℎ

]

[[[[

[

ℎ1,1

⋅ ⋅ ⋅ ℎ1,𝑀𝑝

.

.

....

ℎ𝑁ℎ ,1

⋅ ⋅ ⋅ ℎ𝑁ℎ ,𝑀𝑝

]]]]

]

[[[[[[[

[

𝜔1

𝜔2

.

.

.

𝜔𝑀𝑝

]]]]]]]

]

,

(18)


𝐺 = 200, Size = 30, Code𝐿 = 10, parameters definitioninput vector 𝑋

𝑘= [𝑞hd(𝑘), 𝑞kd(𝑘)]

𝑇, the length of optimized parameters Len = 8initialize population 𝐸 = round(rand(Size, Siez ∗ Code𝐿))for 𝑔 = 1, 2, . . . , 𝐺 dofor 𝑠 = 1, 2, . . . , Size do

for 𝑙 = 1, 2, . . . , Len do𝐹(𝑠, 𝑙) = (Max(𝑙) −Min(𝑙)) ∗ Code(𝑗)/1023 +Min(𝑙); 𝐹(𝑠, 𝑙) will be used for fitness

end forend forSelection and reproductionsort the fitness value and obtain the sequence number indexfor 𝑠 = 1, 2, . . . , Size do

Temp𝐸(𝑗𝑗, :) = 𝐸(index(𝑖), :); 𝑗𝑗 = 𝑗𝑗 + 1

end forCrossover and select the probability 𝑝

𝑐= 0.8

for 𝑠 = 1, 2, . . . , Size dotemp = rand

If 𝑝𝑐> temp do

for 𝑡 = 1, 1, . . . ,Num doTemp𝐸(𝑠, 𝑡) = 𝐸(𝑠 + 1, 𝑡)

Temp𝐸(𝑠 + 1, 𝑡) = 𝐸(𝑠, 𝑡)

end forend if

end forMutation and select the probability

𝑝𝑚= 0.001 − [1 : 1 : Size] ∗ (0.001)/Size, temp = randfor 𝑠 = 1, 2, . . . , Size do

for 𝑗 = 1, 2, . . . , Len doif 𝑝𝑚> temp do

if Temp𝐸(𝑠, 𝑗) == 0 doTemp𝐸(𝑠, 𝑗) = 1

elseTemp𝐸(𝑠, 𝑗) = 0

end ifend if

end forend forreplace old generation with new one

end forObtain optimal parameters

Algorithm 1: Optimize SMC with GA. (Notation: 𝑞hd(𝑘) and 𝑞kd(𝑘) represent desired trajectory of hip joint and knee joint, resp.)

where C𝑠is an association vector projected by input vector,

W is the weight vector,H is the matrix of Hash coding,M𝑝is

the number of Hash vector, 𝑁ℎis the number of association

vector, and ℎ𝑖𝑗= 1 represents 𝑖th association unit response to

𝑗th Hash unit.Similar to other neural networks, the weight parameters

should be updated using Least Square Method (LSM). Theupdating process is expressed as follows:

ΔW =𝜂

𝑁ℎ

A𝑠−1

(y𝑠−1

− A𝑇𝑠−1

W𝑠−1) ,

W (𝑘 + 1) = W (𝑘) + ΔW + 𝛼 (W (𝑘 + 1) −W (𝑘)) ,

(19)

where ΔW is the weight vector increment, 𝜂 is the learningrate, A𝑇

𝑠−1= C𝑇𝑠−1

H, ��𝑠−1

is the target output, and 𝛼 is theinertial parameter.

CMACwas originally proposed to be applied into controlproblems by Miller III et al. [43]. The CMAC control loopis usually added to traditional control loops, where thetraditional controller actuates the plant stably and the CMAChelps to improve control preciseness without affecting thetraditional control loop [44, 45]. In other words, the CMACcontrol is usually added as a compensation item of traditionalcontrol method.The algorithm of the hybrid control strategycombining SMC and CMAC is shown as Algorithm 2 inAppendix B.

3.5. Combination of GA Optimization-Based SMC andCMAC Neural Network. Based on discussion above, we cancombine SMC, GA, and CMAC neural network into ahybrid control strategy, which is called SMC GA CMAC.


input vector 𝑠 = [𝑠1, 𝑠2], determine the range of 𝑠 as 𝑠max 1, 𝑠min 1, 𝑠max 2, 𝑠min 2

initialize CMAC, the range of quantization𝑀, storage of association𝑁, storage of memory 𝐶Input quantization:𝑠1= round((𝑠

1− 𝑠min 1) ∗ 𝑀/(𝑠max 1 − 𝑠min 1)), 𝑠1 = round((𝑠

1− 𝑠min 1) ∗ 𝑀/(𝑠max 1 − 𝑠min 1))

Hash coding and obtain output of CMACfor 𝑘 = 1, 2, . . . , 𝐶 do

add1= mod(𝑠

1+ 𝑘,𝑁) + 1, add

2= mod(𝑠

2+ 𝑘,𝑁) + 1

Sum1 = 𝑤1(add1(𝑘), add2(𝑘)), Sum2 = 𝑤

2(add2(𝑘), add2(𝑘))

end forWeight updatefor 𝑖 = 1, 2, . . . , 𝐶

for 𝑗 = 1, 2, . . . , 𝐶 do𝑑𝑤1(𝑖, 𝑗) = 𝜂𝑒

1/𝐶, 𝑑𝑤

2(𝑖, 𝑗) = 𝜂𝑒

2/𝐶

end forend for𝑤1= 𝑤1 1

+ 𝑑𝑤1+ 𝛼(𝑤

1 1− 𝑤1 2), 𝑤2= 𝑤2 1

+ 𝑑𝑤2+ 𝛼(𝑤

2 1− 𝑤2 2)

𝑤1 2

= 𝑤1 1, 𝑤1 1

= 𝑤1, 𝑤2 2

= 𝑤2 1, 𝑤2 1

= 𝑤2

Algorithm 2: The process of CMAC neural network. (Notation: the input is sliding surface and the output is the compensation controlvector.)

Actuation

Position feedback

Motion intent inference

Human-robot interaction

Actual

Command

pHRI

+

Sliding surface SMC

GA

Weight update law

CMAC

GA parametersGA processGA fitness

Initialization

Exoskeleton

Lower limb

U

−

+

+

e(t)

s1(t)

s2(t)

Us

Uc

AGA,HGA, 𝜀GA,KGA

⌊⌈⌋⌉

⌊⌈⌋⌉

de(t)

e(t)

qd1(t)

qd2(t)

⌊⌈⌋⌉q1(t)

q2(t)

d/dt

⌊⌈⌋⌉

∫e(t)

∫

Figure 4: The control diagram of the proposed method for the exoskeleton system.

We can write the proposed control law based on (9) asfollows:

u = K𝑐U𝑐+M (𝑞) q

𝑑− (T𝑑− C (𝑞, ��) q)

+M (𝑞)AGAe +M (𝑞)HGAe + C (𝑞, ��) s

+ 𝜀GA𝜃 (s) + KGAs,

(20)

where U𝑐represents the output of CMAC neural network,

K𝑐∈ 𝑅𝑛×1 is a positive definite matrix, and AGA, HGA, 𝜀GA,

andKGA arematrices optimized using GA.With the reachingcondition (6), the output of CMAC U

𝑐has constraint as the

following:

U𝑐={

{

{

U𝑐, if s𝑇U

𝑐≥ 0,

−U𝑐, if s𝑇U

𝑐< 0.

(21)

For the exoskeleton system, the control diagram is illus-trated as Figure 4 shows. As Figure 4 shows, GA is employedto obtain the optimal parametersAGA, KGA, HGA, and 𝜀GA toconstruct optimized SMC. The CMAC’s input is the slidingsurface and its weight updating is derived from minimizingthe tracking error. The output of the proposed control law isU = U

𝑠+ K𝑐U𝑐, of which U

𝑠is the main output provided

by SMC GA and U𝑐is the compensation output provided by

CMAC.

4. Simulations with the ProposedControl Strategy

In this section, the proposed method is examined throughsimulations. The simulation results, which are from


0 0.5 1 1.5 2−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (s)

Hip

(rad

)

(a) Desired trajectory of hip joint

0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Knee

(rad

)

(b) Desired trajectory of knee joint

Figure 5: The desired joint trajectory of human lower limb movement. The initial posture is in the vertical direction.

application into controlling the swing leg of the exoskeletonusing the proposed algorithms, are presented. As Figure 1shows, the active DOFs are hip joint and knee joint, whilethe ankle joint is passive. Based on (1), the dynamics modelof swing leg can be expressed as

M (𝑞)exo q + C (𝑞, ��)exo q + G (𝑞)exo = T, (22)

where M(𝑞)exo ∈ 𝑅2×2, C(𝑞, ��)exo ∈ 𝑅

2×2, G(𝑞)exo ∈ 𝑅2×1,

q ∈ R2×1, q ∈ R2×1, and T ∈ R2×1. In simulations, the desiredangular position of lower limb joints stems from the CGAdata as Figure 5 shows. The period of the cyclical gait is 2seconds and we will obtain the fitting expression with respectto time

𝑞hip (𝑡) = 3.85 cos (0.330𝑡 + 2.14)

+ 71.6 cos (3.49𝑡 − 1.88)

+ 41.0 cos (4.68𝑡 − 0.3) ,

𝑞knee (𝑡) = 40.9 cos (1.04𝑡 − 0.208)

+ 157 cos (5.82 − 0.047)

+ 82.3 cos (7.49𝑡 − 4.13) ,

(23)

where 𝑞hip(𝑡) and 𝑞knee(𝑡) are the desired angular position ofhip joint and knee joint, respectively.

To investigate the effectiveness and robustness of the pro-posed scheme, two simulation cases are considered: withoutdisturbances (Case One) and with bounded disturbances(Case Two). The external disturbance 𝐷(𝑡) is a function oftime which is assumed to have an upper bound:

𝐷 (𝑡) = 𝑎 sin (𝜋𝑡) , ‖𝑎‖ ≤ 1. (24)

For recording the respective performances, the root meansquare error (RSME) is defined to examine control perfor-mance as follows:

RSME = √

𝑁

∑

𝑘=1

‖𝑒 (𝑘)‖2

𝑁, (25)

where 𝑁 is the size of error vector. We integrate ADAMSand MATLAB/SIMULINK to control the exoskeleton usingthe proposed control strategy, which is shown in Figure 6. AsFigure 6 shows, there are six output variables from ADAMSmodel which are angular position, velocity, and accelerationof knee joint and hip joint of a swing leg while the designedcontroller in MATLAB outputs two control torques intoADAMS model. Figure 6(a) shows the exoskeleton modelin ADAMS and Figure 6(b) shows the control scheme inSIMULINK.Thedesigned controller produces control signalstransferred to ADAMS while the kinematics information ofexoskeleton joints is measured in ADAMS and returned backto MATLAB workspace. Through creating a communica-tion block between MATLAB and ADAMS, the dynamicsmovements in gait cycles are shown in Figure 7, whichillustrates the level ground walking for the lower extremityexoskeleton.

The comparisons between the proposed control schemeand conventional SMC, SMC with CMAC (SMC CMAC)neural network, and optimal SMC with GA (SMC GA)are conducted. The simulated comparisons, containingtracking positions and tracking errors of SMC, SMC GA,SMC CMAC, and SMC GA CMAC in Case One and CaseTwo, are depicted in Figures 8 and 9. As Figure 8 shows,Figures 8(a) and 8(c) represent the joint trajectory trackingof hip joint and knee joint while Figures 8(b) and 8(d)show tracking error comparisons of those two joints using


(a) ADAMS model

ADAMS/controls2010

adams_sub

To workspace 2

Torque

t

To workspace 4ClockTo workspace 3

To workspace 1

Desired

Actual

S-function1

Signal_Input_Final_Exo

S-function

Main_Ctrl_Final_Exo

Memory

Left_Hip_Acc_Output

Left_Hip_Output

Left_Hip_Vel_Output

Left_Knee_Acc_Output

Left_Knee_Output

Left_Knee_Vel_Output

(b) SIMULINK control model

Figure 6: Cosimulation using ADAMS and MATLAB for exoskeleton robot.

Figure 7: ADAMS effect pictures of gait cycles for lower extremity exoskeleton.


Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4H

ip tr

ajec

tory

(rad

)

(a) Hip joint trajectory tracking

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Knee

traj

ecto

ry (r

ad)

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(b) Knee joint trajectory tracking

−5

0

5

10

Erro

r 1 (r

ad)

×10−3

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA-CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(c) Hip joint tracking errors

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Erro

r 2 (r

ad)

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(d) Knee joint tracking errors

Figure 8: The performance comparisons using four methods in Case One.

four kinds of controllers separately. It can be seen that allof controllers can achieve good tracking performance andthe conventional SMC without optimization has the largesttracking errors. Similarly, the angular position tracking andtracking errors comparisons in Case Two are depicted inFigures 9(a)–9(d). As Figure 9 shows, the desired jointangular trajectory also can be tracked well. To evaluatethe control performances of Case One and Case Two,RSME comparisons using four controllers are depicted in

Figures 10 and 11. Figure 10(a) gives RSME of two jointtracking errors in Case One while Figure 10(b) describesthat in Case Two. In two cases, the performance sequencefromworse to better should be SMC, SMC CMAC, SMC GA,and SMC GA CMAC. Figure 11(a) illustrates the RSMEcomparison of hip joint while Figure 11(b) illustrates that ofknee joint. In Figures 10 and 11, the RSME do not changemuch; hence the proposed control strategy still works whenthere exists external disturbance.


−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4H

ip tr

ajec

tory

(rad

)

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(a) Hip joint trajectory tracking

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Knee

traj

ecto

ry (r

ad)

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(b) Knee joint trajectory tracking

−5

0

5

10

Erro

r 1 (r

ad)

×10−3

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA_CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(c) Hip joint tracking errors

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Erro

r 2 (r

ad)

Time (s)

SMCSMC_GA

SMC_CMACSMC_GA-CMAC

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(d) Knee joint tracking errors

Figure 9: The performance comparisons using four methods in Case Two.

If the RSME is treated as a benchmark, the improvementpercent (IMP) of performance with four kinds of controllersin Case One and Case Two is displayed in Tables 1 and 2. Asthe two tables show, the proposed control strategy will gainthe highest improvement percentages of 69.4% and 76.8% forhip joint and knee joint separately in Case One while theychange to be 68.1% and 76.8% in Case Two. Tables 1 and 2illustrate that the SMC GA is inferior to SMC GA CMACbut has better performance than SMC CMAC, while

the SMC CMAC is superior to SMC. Therefore, theproposed control strategy is robust and effective whether theexoskeleton system dynamics suffer from bounded externaldisturbance or not.

5. Conclusions

For lower limb assistive exoskeletons, precise positioncontrol is very important for the human-exoskeleton


SMC SMC_GA SMC_CMAC SMC_GA_CMAC0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01Er

ror (

rad)

RSME1RSME2

(a) RSME comparisons

RSME1RSME2


0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Erro

r (ra

d)

(b) RSME comparisons

Figure 10: RSME comparisons in Case One (without disturbance) and Case Two (with bounded disturbance), respectively. RSME1 is for thehip joint while RSME2 is for the knee joint.


0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

RSM

E (r

ad)

No-disDis

×10−3

(a) RSME1 comparison in two cases


0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

RSM

E (r

ad)

No-disDis

(b) RSME1 comparison in two cases

Figure 11: RSME1 and RSME2 comparisons in two cases. Icon “No-dis” means Case One and “Dis” means Case Two, respectively.

Table 1: Accuracy improvement comparison (Case One).

Control methods IMP for hip joint (%) IMP for knee joint (%)SMC 0 0SMC GA 57.1% 74.5%SMC CMAC 24.5% 37.4%SMC GA CMAC 69.4% 76.8%

collaboration. In this paper, a hybrid position controlstrategy SMC GA CMAC is proposed to follow humanlimb joints trajectory for the exoskeleton. GA is used to find

Table 2: Accuracy improvement comparison (Case Two).

Control methods IMP for hip joint (%) IMP for knee joint (%)SMC 0 0SMC GA 55.3% 75.2%SMC CMAC 17.4% 31.7%SMC GA CMAC 68.1% 76.8%

the optimal structure of SMC and CMAC neural networkis implemented as the compensation to improve trackingperformance. The proposed SMC GA CMAC control


strategy is proven to be stable with Lyapunov function andfeatures better tracking performance compared with SMC,SMC GA, and SMC CMAC.The proposed control algorithmhas guaranteed the requirement for high accuracy of positioncontrol for robotic manipulators suffering from dynamicsuncertainties. The hybrid control strategy SMC GA CMACis more suitable to control the exoskeleton to follow humanmotion intent under the occurrence of uncertainties. Inaddition, the proposed method will be investigated andexplored in the real exoskeleton prototype in the near future.

Future study will be focused on the optimization ofCMAC to overcome its drawbacks because of the binary inputmapping character, which can be addressed by intelligentapproaches such as fuzzy logic in cosimulation. The humanmotion intent estimation is also a crucial challenge, whichwill be investigated using machine learning methods.

Appendix

A. SMC Optimization Using GA

See Algorithm 1.

B. CMAC Neural Network Compensation

See Algorithm 2.

Competing Interests

The authors declare that they have no competing interests.

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Download - Robust Sliding Mode Control Based on GA Optimization and ......combining SMC and CMAC is shown as Algorithm 2 in AppendixB. 3.5. Combination of GA Optimization-Based SMC and CMACNeuralNetwork

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